The Stacks project

Remark 108.5.9 (Numerical invariants). Let $f : X \to B$ and $\mathcal{F}$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_ i \in D(\mathcal{O}_ X)$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that we have a morphism

\[ \mathrm{Quot}_{\mathcal{F}/X/B} \longrightarrow \mathcal{C}\! \mathit{oh}_{X/B} \]

which sends the element $\mathcal{F}_ T \to \mathcal{Q}$ of $\mathrm{Quot}_{\mathcal{F}/X/B}(T)$ to the object $\mathcal{Q}$ of $\mathcal{C}\! \mathit{oh}_{X/B}$ over $T$, see proof of Quot, Proposition 99.8.4. Hence we can form the fibre product diagram

\[ \xymatrix{ \mathrm{Quot}^ P_{\mathcal{F}/X/B} \ar[r] \ar[d] & \mathcal{C}\! \mathit{oh}^ P_{X/B} \ar[d] \\ \mathrm{Quot}_{\mathcal{F}/X/B} \ar[r] & \mathcal{C}\! \mathit{oh}_{X/B} } \]

This is the defining diagram for the algebraic space in the upper left corner. The left vertical arrow is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \mathbf{Z}$ and $E_ i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_ X$-module $\mathcal{L}$ (in the last case we sometimes use the notation $\mathrm{Quot}^{P, \mathcal{L}}_{\mathcal{F}/X/B}$). See Situation 108.4.7 and Lemmas 108.4.8 and 108.4.9 and Example 108.4.10.

Comments (4)

Comment #2815 by on

The maths display after "Hence we can form the fibre product diagram" is broken - I get a parse error. The source gives

\Quotfunctor^P_{\mathcal{F}/X/B} \ar[r] \ar[d] &

but the error message says the error is 'at or near'

{\rm Quot}^P_{\mathcal{F}/X/B} \ar[r] \ar[d] &

Comment #2816 by on

@2815: Indeed, the diagram is broken. This appears to be an issue with XyJax. Specifically, it appears XyJax is not very happy with {\rm Quot}. In general, XyJax does not like this {\rm ...} for making roman text in its diagrams. See Tag 09BX for another instance.

Comment #2817 by on

@2815: The difference between what you see in the source and the XyJax error is because we are replacing some macros. So it's not the difference between these that causes the error, but rather what Raymond explains.

Because comments are subject to the same conversion, your comment seems to make little sense to anyone but me. But there are actually 2 different lines of source code there.

Comment #2918 by on

Hi David, Raymond, and Pieter. I am going to leave this for now. Please look in the pdf until this gets fixed in the new website.

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View Remark 108.5.9 as pdf